and \(k\)  is the ratio of relativistic proton to relativistic electron energy, \(V\)  is the source volume, \(C_{12}\)  is a constant which depends on the spectral index and frequency cutoffs [1]}, \(L_{rad}\)  is the total radio luminosity, \(\phi \)  is the volume filling factor, \(z\)  is the source redshift, \(D_{Mpc}\)  is the luminosity distance of the source in Mpc, \(S_0\)  is the flux density in Jy at a fiducial frequency \(\nu _0\)  in Hz, \(\alpha \)  is the spectral index, and \(\nu _u\)  and \(\nu _l\)  are the upper and lower frequency cutoffs in Hz.
Using the following assumptions:
